Respuesta :
Answer:
[tex]\dfrac{d\theta}{dt}= 0.293\ rad/s[/tex]
Explanation:
horizontal distance traveled by the bird is equal to be x
given,
height of the bird above his head = 25 m
angle between them is equal to θ
[tex]\dfrac{dx}{dt} = 9 m/s[/tex]
d² = x² + y²
[tex]\theta = tan^{-1}{\dfrac{x}{25}}[/tex]
[tex]\dfrac{d\theta}{dx}= \dfrac{1}{1+(\dfrac{x}{25})^2}\times(\dfrac{1}{25})[/tex]
[tex]\dfrac{d\theta}{dx}= \dfrac{25}{x^2+625}[/tex]
now,
[tex]\dfrac{d\theta}{dt}= \dfrac{d\theta}{dx}\times \dfrac{dx}{dt}[/tex]
[tex]\dfrac{d\theta}{dt}= \dfrac{25}{x^2+625}\times 9[/tex]
[tex]\dfrac{d\theta}{dt}= \dfrac{225}{x^2+625}[/tex]
at x = 12 m
[tex]\dfrac{d\theta}{dt}= \dfrac{225}{12^2+625}[/tex]
[tex]\dfrac{d\theta}{dt}= 0.293\ rad/s[/tex]
The speed of the bird when the horizontal distance between you and the bird is 12 meters is 0.2926 rad/sec.
What is Velocity?
Velocity is the rate of change of position of the object with respect to time.
[tex]v=\dfrac{dx}{dt}[/tex]
We know that the vertical distance between you the bird is 25 m, while the speed of the bird is 9 m/s. Therefore, the change θ in the horizontal distance between you and the bird can be given as,
[tex]\dfrac{d \theta}{dt}[/tex]
Further, the change can be written as
[tex]\dfrac{d \theta}{dt} = \dfrac{d \theta}{dx} \times \dfrac{d x}{dt}[/tex]
We know that the angle between the bird and you can be written as θ,
[tex]\theta = tan^{-1}\ \dfrac{x}{25}[/tex]
Differentiating the angle with respect to x,
[tex]\dfrac{d\theta}{dx} = \dfrac{1}{1+(\dfrac{x}{25})^2} \times \dfrac{1}{25}\\\\\dfrac{d\theta}{dx}=\dfrac{25}{x^2+625}[/tex]
As it is given that the velocity of the bird is 9 m/s, we know that the velocity can be written as the rate of change of position of the object with respect to time. therefore,
[tex]v=9{\rm\ m/s}\\\\\dfrac{dx}{dt} = 9\rm\ m/s[/tex]
Now, the change θ in the horizontal distance between you and the bird can be written as,
[tex]\dfrac{d \theta}{dt} = \dfrac{d \theta}{dx} \times \dfrac{d x}{dt}\\\\\dfrac{d \theta}{dt} = \dfrac{25}{x^2+625} \times 9\\\\\dfrac{d \theta}{dt} = \dfrac{225}{x^2+625}[/tex]
As we need to know the change θ in the horizontal distance between you and the bird when the horizontal distance between you and the bird is 12 meters. therefore,
[tex]\dfrac{d \theta}{dt} = \dfrac{225}{x^2+625}\\\\\dfrac{d \theta}{dt} = \dfrac{225}{(12)^2+625}\\\\\dfrac{d \theta}{dt} = \dfrac{225}{769}\\\\\dfrac{d \theta}{dt} =0.2926\rm\ rad/sec[/tex]
Hence, the speed of the bird when the horizontal distance between you and the bird is 12 meters is 0.2926 rad/sec.
Learn more about Velocity:
https://brainly.com/question/862972
